Analytic capacity and removable sets

In this chapter we shall discuss a classical problem in complex analysis and its relations to the rectifiability of sets in the complex plane C. The problem is the following: which compact sets E ⊃ C are removable for bounded analytic functions in the following sense?

(19.1) If U is an open set in C containing E and f: U\E → C is a bounded analytic function, then f has an analytic extension to U.

This problem has been studied for almost a century, but a geometric characterization of such removable sets is still lacking. We shall prove some partial results and discuss some other results and conjectures. For many different function classes a complete solution has been given in terms of Hausdorff measures or capacities. For example, if the boundedness is replaced by the Holder continuity with exponent α, 0 < α < 1, then the necessary and sufficient condition for the removability of E is that H1+α(E) = 0, see Exercise 4, Dolzenko [1] and Uy [2], and for the corresponding question for harmonic functions Carleson [1]. Král [1] proved that for the analytic BMO functions the removable sets E are characterized by the condition H1(E) = 0. The problem (19.1) is more delicate, because the metric size is not the only thing that matters; the rectifiability structure also seems to be essential as we shall see.

Ahlfors [1] introduced a set function γ, called analytic capacity, whose null-sets are exactly the removable sets of (19.1).

This is a book on geometric measure theory. The main theme is the study of the geometric structure of general Borel sets and Borel measures in the euclidean n-space Rn. There will be emphasis on “small irregular” sets having Lebesgue measure zero but being quite different from smooth curves and surfaces. Examples are Cantor-type sets, nonrectifiable curves having tangent nowhere, etc., in short, sets to which the general descriptive term fractal applies. An abundance of such sets comes from dynamical systems: Julia-sets for rational functions of one complex variable, etc. Very general curve – and surface-like objects are also studied extensively. These are rectifiable sets and measures. They include smooth curves and surfaces and share many of their geometric properties when interpreted in a measure-theoretic sense. They form an optimal class possessing such properties.

Many of the basic ideas developed here originate in the pioneering work done by Besicovitch [1], [4] and [5], by Federer [1], by Marstrand [1] and by Preiss [4]. Besicovitch laid down the foundations of geometric measure theory by describing to an amazing extent the structure of the subsets of the plane having finite one-dimensional Hausdorff measure (i.e. length). Federer extended Besicovitch's work to m-dimensional subsets of Rn, m being an integer, and Marstrand analysed general fractals in the plane whose Hausdorff dimension need not be an integer. Preiss solved one of the most long-standing fundamental open problems, introducing and using effectively tangent measures.

Good introductory texts to the mathematical theory of fractals are the books of Edgar [1] and of Falconer [4], [16]. Closest to this text is Falconer [4].